Problem: $\dfrac{ -5l - 3m }{ 4 } = \dfrac{ -8l + 5n }{ 7 }$ Solve for $l$.
Answer: Multiply both sides by the left denominator. $\dfrac{ -5l - 3m }{ {4} } = \dfrac{ -8l + 5n }{ 7 }$ ${4} \cdot \dfrac{ -5l - 3m }{ {4} } = {4} \cdot \dfrac{ -8l + 5n }{ 7 }$ $-5l - 3m = {4} \cdot \dfrac { -8l + 5n }{ 7 }$ Multiply both sides by the right denominator. $-5l - 3m = 4 \cdot \dfrac{ -8l + 5n }{ {7} }$ ${7} \cdot \left( -5l - 3m \right) = {7} \cdot 4 \cdot \dfrac{ -8l + 5n }{ {7} }$ ${7} \cdot \left( -5l - 3m \right) = 4 \cdot \left( -8l + 5n \right)$ Distribute both sides ${7} \cdot \left( -5l - 3m \right) = {4} \cdot \left( -8l + 5n \right)$ $-{35}l - {21}m = -{32}l + {20}n$ Combine $l$ terms on the left. $-{35l} - 21m = -{32l} + 20n$ $-{3l} - 21m = 20n$ Move the $m$ term to the right. $-3l - {21m} = 20n$ $-3l = 20n + {21m}$ Isolate $l$ by dividing both sides by its coefficient. $-{3}l = 20n + 21m$ $l = \dfrac{ 20n + 21m }{ -{3} }$ Swap signs so the denominator isn't negative. $l = \dfrac{ -{20}n - {21}m }{ {3} }$